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Quantum simulation of elastic wave equations via Schrödingerisation

Shi Jin, Chundan Zhang

TL;DR

This work tackles the quantum simulation of elastic wave propagation by applying Schrödingerisation to two core formulations: the symmetric matrix form (SMF) velocity-stress equations in isotropic media and a hyperbolic first-order system for displacement, including variable-coefficient media with a staggered-grid discretization. By mapping these linear PDEs to higher-dimensional unitary Hamiltonian systems, the authors derive quantum algorithms based on spectral and central-difference spatial discretizations and provide rigorous complexity analyses that reveal exponential advantages in the spatial dimension under suitable discretizations. They present detailed constructions of the Schrödingerisation operators, recovery procedures for original fields, and complexity bounds, supported by comprehensive numerical experiments that validate the approach against classical solutions. The results offer a principled pathway toward quantum-accelerated, large-scale elastic-wave simulations with potential impact in geophysics and materials science, while outlining future work on more general boundary-value problems and practical quantum-circuit implementations.

Abstract

In this paper we study quantum simulation algorithms on the elastic wave equations using the Schrödingerisation method. The Schrödingerisation method transforms any linear PDEs into a system of Schrödinger-type PDEs -with unitary evolution-using the warped phase transformation that maps the equations in one higher dimension. This makes them suitable for quantum simulations. We expore the application in two forms of the elastic wave equations. For the velocity-stress equation in isotropic media, we explore the symmetric matrix form under the external forcing via Schrödingerisation combined with spectral method. For problems with variable medium parameters, we apply Schrödingerisation method based on the staggered grid method to simulate velocity and stress fields, and give the complexity estimates. For the wave displacement equation, we transform it into a hyperbolic system and apply the Schrödingerisation method, which is then discretized by the spectral method and central difference scheme. Details of the quantum algorithms will be provided, along with the complexity analysis which demontrate exponential quantum advantage in space dimensin over the classical algorithms.

Quantum simulation of elastic wave equations via Schrödingerisation

TL;DR

This work tackles the quantum simulation of elastic wave propagation by applying Schrödingerisation to two core formulations: the symmetric matrix form (SMF) velocity-stress equations in isotropic media and a hyperbolic first-order system for displacement, including variable-coefficient media with a staggered-grid discretization. By mapping these linear PDEs to higher-dimensional unitary Hamiltonian systems, the authors derive quantum algorithms based on spectral and central-difference spatial discretizations and provide rigorous complexity analyses that reveal exponential advantages in the spatial dimension under suitable discretizations. They present detailed constructions of the Schrödingerisation operators, recovery procedures for original fields, and complexity bounds, supported by comprehensive numerical experiments that validate the approach against classical solutions. The results offer a principled pathway toward quantum-accelerated, large-scale elastic-wave simulations with potential impact in geophysics and materials science, while outlining future work on more general boundary-value problems and practical quantum-circuit implementations.

Abstract

In this paper we study quantum simulation algorithms on the elastic wave equations using the Schrödingerisation method. The Schrödingerisation method transforms any linear PDEs into a system of Schrödinger-type PDEs -with unitary evolution-using the warped phase transformation that maps the equations in one higher dimension. This makes them suitable for quantum simulations. We expore the application in two forms of the elastic wave equations. For the velocity-stress equation in isotropic media, we explore the symmetric matrix form under the external forcing via Schrödingerisation combined with spectral method. For problems with variable medium parameters, we apply Schrödingerisation method based on the staggered grid method to simulate velocity and stress fields, and give the complexity estimates. For the wave displacement equation, we transform it into a hyperbolic system and apply the Schrödingerisation method, which is then discretized by the spectral method and central difference scheme. Details of the quantum algorithms will be provided, along with the complexity analysis which demontrate exponential quantum advantage in space dimensin over the classical algorithms.

Paper Structure

This paper contains 26 sections, 6 theorems, 106 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. A review of elastic wave equations
  3. The first-order form of the elastic wave equation
  4. The Symmetric Matrix Form Framework for the Elastic Wave Equation
  5. The hyperbolic system of wave displacement equation
  6. A review of the Schr$\ddot{\text{o}}$dingerisation methed
  7. The Schr$\ddot{\text{o}}$dingerisation method
  8. Recovery of solutions
  9. The complexity
  10. Quantum simulation of the first-order velosity-stress equation
  11. Quantum simulation of SMF of elastic wave equation \ref{['v-s_SMF']} by spectral method
  12. Schr$\ddot{\text{o}}$dingerisation Simulation
  13. Quantum simulation of velocity-stress equation with variable coefficients by staggered grid method
  14. The Schr$\ddot{\text{o}}$dingerisation Simulation
  15. Quantum simulation of elastic wave displacement equations in isotropic media
  16. ...and 11 more sections

Key Result

Lemma 3.1

OptimalHS For evolution time $T$ and error $\epsilon$, a time-independent $s$-sparse Hamiltonian $H$ on $m_H$ qubits subject to $m_e$ bits of precision, has the query complexity with respect to $\tau = s\|H\|_{max} T$ as which achieves known lower bounds additively with an additional primitive gate complexity

Figures (4)

  • Figure 1: Velocity and stress fields under the constant external force at $T=1$.
  • Figure 2: Result at $T=1$ with periodic boundary condition. On the left are the numerical solutions obtained by staggered grid method, in the middle and right are the cross-section solutions at $x^*=5$ and $y^*=5$. The top two rows present the velocity field, and the bottom three rows present the stress field.
  • Figure 3: Results of the hyperbolic system at $T=1$ using the Schr$\ddot{\text{o}}$dingerisation with the spectral method. The first row: using the first row parameters in Tab. \ref{['Tab:parameter']} ; The second row: using the second parameters in Tab. \ref{['Tab:parameter']}.
  • Figure 4: Results of the hyperbolic system at $T=1$ using the Schr$\ddot{\text{o}}$dingerisation with the central difference method. The first row: using the first row parameters in Tab. \ref{['Tab:parameter']} ; The second row: using the second parameters in Tab. \ref{['Tab:parameter']}.

Theorems & Definitions (11)

  • Lemma 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • Theorem 5.3
  • ...and 1 more